3.2.9
Ek orbit ek frozen ellipse hai jo 3D space mein float kar raha hai. Isko completely pin down karne ke liye tumhe chhe sawaalon ke jawaab dene honge:
Kitna bada hai ellipse? → a Kitna squashed hai? → e Kitna tilted hai iska plane reference plane ke relative? → i Kis taraf tilted plane ghoomta hai? → Ω Kis taraf ellipse point karta hai apne plane ke andar? → ω Satellite abhi kahan hai ellipse par? → ν M E
Pehle do (a , e shape describe karte hain. Agle teen (i , Ω , ω orientation describe karte hain. Aakhri ek (ν position in time describe karta hai. Ye exactly 6 numbers hain — aur position+velocity state ( r , v )
Definition State vector vs. orbital elementsEk satellite ki motion poori tarah se fix hoti hai uski position r v 6 numbers . Orbital elements 6 numbers ka ek alternative set hai jo wohi orbit describe karta hai, lekin ek aisa tarika jise humans geometrically visualize kar sakein aur jo two-body problem mein zyaadatar constant rehte hain .
POINT kya hai? ( r , v ) chhe mein se paanch (a , e , i , Ω , ω constant hote hain — sirf ν
Intuition Reference ki zaroorat kyun hai"Tilted" ka koi matlab nahi jab tak tum yeh na bolo ki kisse relative tilted hai . Hum choose karte hain:
ek reference plane (Earth ke liye: equatorial plane),
us plane mein ek reference direction : vernal equinox direction Υ ^
aur orbit ka ascending node — wo point jahan satellite reference plane ko upar ki taraf cross karta hai (south→north).
Definition Semi-major axisa size set karta hai aur Kepler ke 3rd law se, period aur total energy bhi.
e squashed hai, bound orbits ke liye 0 ≤ e < 1
e = 0 e → 1 e = 1 e > 1
Worked example Peri/apo sea , e r p = 7000 r a = 9000
a = 2 r p + r a = 8000 Kyun? Do extreme points poora major axis 2 a e = r a + r p r a − r p = 16000 2000 = 0.125 Kyun? Numerator 2 a e 2 a e
i 0 ≤ i ≤ 18 0 ∘
i = 0 prograde (Earth ke spin ke same direction mein orbit).i = 9 0 ∘ polar (dono poles ke upar se guzarta hai).i > 9 0 ∘ retrograde .
Ω Ω reference plane mein measure kiya hua, vernal equinox Υ ^ ascending node tak (upar ki taraf crossing point). Ye batata hai ki tilted plane z
Definition Argument of periapsisω orbital plane mein measure kiya hua, ascending node se periapsis (closest approach) tak, motion ki direction mein.
ν ek maatra element hai jo two-body orbit mein time ke saath badalta hai.
Recall Feynman: ek 12-saal ke bacche ko explain karoSocho ek oval racetrack ko space mein tilte hue dinner plate par draw kar rahe ho.
a e i Ω ω par oval kis taraf point karta hai.ν
Mnemonic Order = Shape → Orientation → Position"A Sweet Idea Of Wonderful Vistas" → a e i Ω ω ν 2 shape, 3 orientation, 1 position.
Common mistake "Inclination batata hai ki plane kis taraf rotate hua hai."Kyun sahi lagta hai: i Ω Fix: i kitna steep hai (equator se angle). Ω kis compass direction mein face karta hai (z i = 9 0 ∘ Ω
a Kyun sahi lagta hai: circle ke liye, r = a Fix: a major axis ka aadha hai, ek constant; actual distance r r p r a e = 0 r = a
a e Kyun sahi lagta hai: speed clearly e Fix: ε = − μ /2 a sirf a depend karti hai. Eccentricity orbit ke around kinetic↔potential energy ko reshuffle karta hai lekin total fixed rehta hai.
ω ν Kyun sahi lagta hai: dono motion ki direction mein in-plane angles hain.
Fix: ω ν u = ω + ν
Kaun sa element orbital period aur energy set karta hai? Semi-major axis
a (kyunki
ε = − μ /2 a aur
T = 2 π a 3 / μ ).
Kaun sa orbital element two-body orbit mein time ke saath badalta hai? Sirf true anomaly ν
Eccentricity ko words mein define karo aur 0/1 Ellipse kitna squashed hai; e = 0 e → 1
r p , r a a e a = ( r p + r a ) /2 e = ( r a − r p ) / ( r a + r p )
Inclination i Orbital plane aur equatorial reference plane ke beech ka angle.
i = 9 0 ∘ Polar orbit (poles ke upar se guzarta hai).
Ascending node kya hai? Wo point jahan satellite reference plane ko south→north (upar ki taraf) cross karta hai.
RAAN Ω Reference plane mein vernal equinox se ascending node tak ka angle (plane ka z
Argument of periapsis ω Motion ke saath in-plane angle ascending node se periapsis tak.
Periapsis/apoapsis radii derive karo. r p = a ( 1 − e ) r a = a ( 1 + e ) 2 a
Kaun sa vector periapsis ki taraf point karta hai aur iski magnitude e State vector se inclination kaise nikalte hain? cos i = h z /∣ h ∣ jahan
h = r × v .
Velocity sign ν r ⋅ v > 0 → outbound (
0 < ν < 18 0 ∘ );
< 0 → inbound.
Elements ke saath orbit (polar) equation? r = a ( 1 − e 2 ) / ( 1 + e cos ν )
Chhe elements ka 3-2-1 split? 2 shape (a , e i , Ω , ω ν
Vis-viva equation — a , r v ε = − μ /2 a Kepler's Laws — third law a T Angular momentum in orbits — h i Ω Eccentricity vector — e ω State vector to orbital elements conversion — in meanings ka use karne wala full algorithm.Orbital perturbations — real (non-ideal) world mein Ω , ω
State vector r and v
Orbital elements
Reference frame
